Let the area of the region \(\left\{(x, y): 2 y \leq x^2+3, y+|x| \leq 3, y \geqslant|x-1|\right\}\) be A. Then 6 A is equal to :

• A

  16
• B

  12
• Correct answer
  C

  14
• D

  18

Solution

$A \Rightarrow$ Rectangle ABDE - Area of region EDC $\begin{aligned} & A \Rightarrow 4-2 \int_0^1(3-x)-\left(\frac{x^2+3}{2}\right) d x \\ & A \Rightarrow 4-2\left\{3 x-\frac{x^2}{2}-\frac{x^3}{6}-\frac{3}{2} x\right\}_0^1 \\ & A \Rightarrow 4-2\left\{3-\frac{1}{2}-\frac{1}{6}-\frac{3}{2}\right\}=\frac{7}{3} \end{aligned}$ So $6 \mathrm{~A}=14$

Hence, the correct answer is option: 'C'.